Linear approximation techniques offer an over-approximation zone of every layer's function in essence. DeepPoly [1]} first discovers that minimizing the over-approximation zone can give rise to a more precise certified robustness bound. However, empirical analysis in Newise [2]} shows that the linear constraints that producing the minimal over-approximation zone could not lead to the most precise robustness result all the time, which motivates us to find the relation between minimizing the over-approximation zone and Tightest robustness bound. Therefore, we introduce the notion of Neuron-wise Tightest Linear Bounds from Newise.
[Neuron-wise Tightest]

upper linear approximation \(u_i^k:\times _{q\in [n_{k-1}]}[l_q^{k-1},u_q^{k-1}]\rightarrow \mathbb {R}\) , lower linear approximation \(l_i^k:\times _{q\in [n_{k-1}]}[l_q^{k-1},u_q^{k-1}]\rightarrow \mathbb {R}\)  is Neuron-wise Tightest iff \(D^{k-1}=\times _{q\in [n_{k-1}]}[l_q^{k-1},u_q^{k-1}], \forall j\in [n_{k+1}]\) ,\(\iint _{D^{k-1}}u_i^k(x) dx\)  and \(\iint _{D^{k-1}}-l_i^k(x) dx\)  are minimal

We discover Neuron-wise Tightest linear bounds can keep the Tightest through one-layer propagation(proved in appendix). Therefore, Neuron-wise Tightest can explain why narrowing the over-approximation zone can increase certified robustness bounds. Furthermore, Newise defines the notion of Network-wise Tightest, that is, Network-wise Tightest linear bounds can keep the Tightest through the whole network propagation. Furthermore, Newise provides the Network-wise Tightest linear bounds for monotonous networks, such as non-negative networks [2]}.
[Network-wise Tightest]

global upper linear approximation \(U_i^K:\times _{q\in [n_{0}]}[x_{0,q}-\epsilon ,x_{0,q}+\epsilon ]\rightarrow \mathbb {R}\) , global lower linear approximation \(L_i:\times _{q\in [n_{0}]}[x_{0,q}-\epsilon ,x_{0,q}+\epsilon ]\rightarrow \mathbb {R}\)  is network-wise Tightest iff for any different global linear constraints \(\hat{U}_i^K, \hat{L}_i^K\) ,
\(max U_i^K(x)\le max\hat{U}_i^K(x), \,\,min L_i^K(x)\ge min\hat{L}_i^K(x), x\in \mathbb {B}_p(x_0,\epsilon )\) 
